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Continuous cluster categories II: Continuous clustertilted categories. (with Gordana Todorov)

Revising
Oct 11, 2014



Since CCCI has now appeared, we are working to finish part II. This paper starts with a review of the definitions and results of CCC I. This revised version has complete proofs of the main theorems, namely (1) the rational and continuous clustertilted categories are abelian and (2) they are isomorphic to categories of string modules over the Jacobian algebra of an infinite quiver with potential. Further revisions are needed to incorporate the precise descritption of the triangulation of the continuous categories given by the continuous Frobenius category.


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Periodic trees and propictures. (with Moses Kim, Gordana Todorov, Jerzy Weyman)

Preliminary version
March, 2018


This is a continuation of the paper Periodic trees and semiinvariants. The semiinvariant picture for an affine quiver is a inverse limit of pictures for an inverse system of finitely presented groups. The walls of the compartments separating the periodic functions for each period tree are separated by domains of these semiinvariants. The propicture group with infinite relations are extended to affine quivers with potential.


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Cluster morphism category as cubical category. (with Gordana Todorov)

Later


Using a result of Speyer and Thomas and Lemma 1.3.1 from ``Picture groups and maximal green sequences'' we show that, for modulated quivers of finite type, the classifying space of the cluster morphism category is a cubical category whose geometric realization is a nonpositively curved cube complex for quivers of finite type. For tame infinite type, it is not npc and, therefore, not a K(pi,1). This gives another proof that the space X(Q) defined in the paper "Picture groups of finite type and cohomology in type A_n" is a K(pi,1).


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Picture groups and maximal green sequences. (with Gordana Todorov)

IT14.v140923


It is easy to see that there is a monomorphism from the set of maximal green sequences for an acyclic quiver into the set of positive expressions for the coxeter element of the corresponding picture group. This paper show that, in finite type, this mapping is a bijection. In the new version, the statement of Lemma 1.3.1 is modified.

